Is it possible to animate a fire emitter/domain combo using armature and weightpaint - and if it is, how do I do that?
I see two ways to do that. You can either parent the domain directly to a single bone or use the "Child of" constraint.

Parent to a single bone
First select the Domain object object mode, then in pose mode select a parent bone of your character. Then press ctrl+p and select bone. Now the domain object moves with your parent bone.
Use a Child of constraint
In object mode select your domain object. Then in the properties panel go to the object constrains tab and add a new "Child of" constraint. Under "Target" select your armature and under "Bone" select the parent bone of your character. Click "set inverse" to reset the transformations. Now your domain object moves with your parent bone.

I'm not quite sure if this is recommended to do. As this will probably influence the simulation as the reference frame is moved. Not sure if this will cause problems.